Statistical method of averaging

[1] 112.2274

Fake Data and the average as an estimate

  • The mean value summarises the path price into a single number
  • But how accurate is this algorithm?
  • The textbook answer is given in terms of the standard error

Standard error function

[1] 0.98525
  • Here averaging is the .content-box-red[algorithm], while the standard error provides the .content-box-blue[inference] of the algorithm’s accuracy.

Explanation

  • It is a surprising, and crucial, aspect of statistical theory that the same data that supplies an estimate can also assess its accuracy.
  • Strictly speaking Inference concerns more than accuracy: recall that algorithms say what the statistician does while inference says why she does it.

Algorithms and Inference

Algorithms and regression

.panelset[ .panel[.panel-name[Least squares algorithm for linear regression] - The least squares estimator is a popular algorithm for estimating a linear regression - The algorithm fits the data by least squares, by minimising the sum of squared deviations over all choices of the model parameters. - Consider the following fake relationship between the price and some market factor

Least squares algorithm

Lowess algorithm for localised regression

  • Lowess is a modern computer based algorithm which works by moving its attention along the x-axis, fitting local polynomial curves of differing degrees to nearby (x,y) coordinates.
  • The fitted estimate above has a similar linear regression as the least squares algorithm in the middle of the data but for higher values of the factor has a much steeper curve.

Hypothesis testing

Two sample t-test inference

  • We obtain t= 5.07 which would classically be considered very strong evidence that the apparent difference is genuine; in standard terminology, “with two-sided significance level 0.0000003.”
  • A small significance level (or “p-value”) is a statement of statistical surprise: something very unusual has happened if in fact there is no difference in returns of the value factor in up and down markets
  • We are less surprised by t=5.07 if this comparison of mean returns is just one candidate out of thousands that might have produced “interesting” results.
  • For example if I take a different set sample period and run the test again, or I split the sample into 5 sub-samples and run the test again.

Traditional hypothesis testing and false discovery theory

  • A primary goal of empirical research is to identify the variables involved in a phenomenon
  • The identification of these variables is a prerequisite to the formulation of a theory
  • In classical statistics (e.g., Econometric), the significance of variables is established through p values
  • p values suffer from multiple flaws, which have led to the acknowledgment that most discoveries in finance are false
  • False discovery rate theory is an impressive advance in statistical inference, incorporating Bayesian ,frequentist, and empirical Bayesian elements.
  • It was a necessary advance in a scientific world where computer-based technology routinely presents thousands of comparisons to be evaluated at once.

Simulated example of false discovery problem

set.seed(1235)
res<-tibble(testno=1:20, pvalue=1)
for (i in 1:20) {
  Up=rnorm(1000)
  NoUp=rnorm(1000)
  res[i,2]<-t.test(Up,NoUp)['p.value']
}
  • This code fakes the return up versus down question earlier, where the ground truth is that there is no relationship.
  • This implies that our significant results, when using a t-test, must be fake
  • We repeat the test 20 times using a simple for loop.
  • Arranging the results in ascending order of p-values.
  • With enough hypothesis test we will always discovery a fake results.
  • In financial machine learning, where we have to work with high-dimensional data with many features, 1000’s of comparisons are common.
---
title: "Computational notebook for PhD Seminar"
output: html_notebook
---
```{r set-up}
knitr::opts_chunk$set(echo = FALSE)
pacman::p_load("xaringanExtra","knitr","kableExtra","fontawesome","tidyverse","xaringanthemer")
```



# Statistical method of averaging

- Consider the simplistic but most popular measure of averaging 
- Suppose we observe a set of the price change path phenomenon of a stock over t periods $x_1, x_2,...,x_t$.

```{r random-walk-simulator, fig.height=4, fig.width=8}
set.seed(123)
S1=100; t=100; Price=vector(length = t)
noise=runif(t,min = -5,max = 5)
for (tt in 1:t) {
  if (tt==1) {
    Price[tt]=S1
  }
  else{
  Price[tt] = Price[tt-1] + noise[tt]
  }
}
sum(Price)/length(Price)
```

- The averaging function highlighted above is an algorithm and is represented by the red line in the plot opposite

## Fake Data and the average as an estimate

```{r echo=FALSE, fig.height=5, fig.width=7}
price_mean=sum(Price)/length(Price)
Price %>% plot(type="l") %>% abline(h=price_mean, col="red")
```

- The mean value summarises the path price into a single number
- But how accurate is this algorithm?
- The textbook answer is given in terms of the *standard error*

## Standard error function

```{r, warning=FALSE}
standard_error<- function(x){
  dx=(x-mean(x))^2
  denom=length(x)*(length(x)-1)
  sumsq=sum(dx)/denom
  sqrt(sumsq)
}
standard_error(Price)
```
- Here averaging is the .content-box-red[algorithm], while the standard error provides the .content-box-blue[inference] of the algorithm's accuracy.

## Explanation

- It is a surprising, and crucial, aspect of statistical theory that the same data that supplies an estimate can also assess its accuracy. 
- Strictly speaking *Inference* concerns more than accuracy: recall that algorithms say what the statistician does while inference says why she does it.

# Algorithms and Inference
- Of course, the `standard_error()` function defined previously is itself an algorithm, which could be (and is) subject to further inferential analysis concerning its accuracy 

- .content-box-green[The point is that the algorithm comes first and the inference follows at a second level of statistical consideration.]

- .content-box-yellow[In practice this means that algorithmic invention is a more free-wheeling and adventurous enterprise]

- .content-box-grey[In contrast inference playing catch-up as it strives to assess the accuracy, good or bad, of some hot new algorithmic methodology.]


# Algorithms and regression
.panelset[
.panel[.panel-name[Least squares algorithm for linear regression]
- The least squares estimator is a popular algorithm for estimating a linear regression
- The algorithm fits the data by *least squares*, by minimising the sum of squared deviations over all choices of the model parameters.
- Consider the following fake relationship between the price and some market factor
```{r fake data}
factor<-Price + runif(t,1,8)^2 #<<
tibble(Price,factor)->df 
```

Least squares algorithm

```{r echo=FALSE}
df %>% ggplot(aes(y=Price,x=factor)) + geom_point() +
  geom_smooth(method = "lm") #<< adds least squares line with standard errors
```

- This code manufactures a positive relationship to the factor plus some noise and then draws the least squares regression line.
- The accuracy of this estimate is given by $\pm$ 2 standard errors
- The appropriate inference of this banded grey area is that this has a 95% chance in including the true expected value of `Price` in an `Up` market.
- This 95% coverage depends on the validity of the linear regression model, which could as easy have been a quadratic relationship

## Lowess algorithm for localised regression

* Lowess is a modern computer based algorithm which works by moving its attention along the x-axis, fitting local polynomial curves of differing degrees to nearby `(x,y)` coordinates. 
- The fitted estimate above has a similar linear regression as the least squares algorithm in the middle of the data but for higher values of the factor has a much steeper curve.

```{r loess}
df %>% ggplot(aes(y=Price,x=factor)) + geom_point() +
  geom_smooth(method = "loess") # 
```


```{r animation code}
set.seed(123)
S1=100; t=100; Price=vector(length = t)
noise=runif(t,min = -5,max = 5)
for (tt in 1:t) {
  if (tt==1) {
    Price[tt]=S1
  }
  else{
    Price[tt] = Price[tt-1] + noise[tt]
  }
}
factor<-Price + runif(t,1,8)^2 # <<

library(animation)
library(tidyverse)
ani.options(nmax = 20, interval = 0.05)
tibble(Price=Price,factor=factor) -> df
  boot.lowess(y =df$Price,x=df$factor, f=2/3,iter = 10,line.col = "red",
              xlab = "factor", ylab = "Price") %>%
  saveGIF(movie.name = "lowess_ani.gif")

```

# Hypothesis testing
```{r}
library(ati)
#  install ati from my github using
# remove.packages('ati')
# .rs.restartR()
# remotes::install_github("barryquinn1/ati")
ati::daily_factors->uk_factors
uk_factors %>% mutate(Bull=if_else(rm>0,"Up","Not Up")) -> uk_factors
meanUP<-mean(uk_factors$hml[uk_factors$Bull=='Up'])
mean_notUP<-mean(uk_factors$hml[uk_factors$Bull=='Not Up'])
```

- There has also been a march of methodology and inference for hypothesis testing rather than estimation
- Consider questions about where value investing is better in a bull or a bear market?
- The plots shows the histogram of these two groups in the uk data.Up markets to seem to have a more positive return then down markets.
- The mean values for up and not up are in fact `r sprintf("%.5f",meanUP)` and `r sprintf("%.5f",mean_notUP)` respectively 
- Is the perceived difference genuine or as some people would say **a statistical fluke**

```{r, fig.width=8}
uk_factors %>% ggplot(aes(x=hml, fill=Bull)) +
  geom_histogram(bins=15) +
  facet_wrap(~Bull)
```

- The classic answer to this question is via a two-sample t-test
$$\frac{\bar{Value_{UP}}-\bar{Value_{Not Up}}}{\hat{sd}}$$
- where $\hat{sd}$ is estimate of the numerators standard deviation

- .content-box-blue[Dividing by sd allows us (under Gaussian assumptions) to compare the observed value of t with a standard **null** distribution, in this case a Student’s t distribution with `r nrow(uk_factors)-1` degrees of freedom.]


## Two sample t-test inference

```{r}
t.test(uk_factors$hml[uk_factors$Bull=='Up'],uk_factors$hml[uk_factors$Bull!='Up'])
```

- We obtain t= 5.07 which would classically be considered very strong evidence that the apparent difference is genuine; in standard terminology, “with two-sided significance level 0.0000003.”
- A small significance level (or “p-value”) is a statement of statistical surprise: something very unusual has happened if in fact there is no difference in returns of the value factor in up and down markets
- We are less surprised by t=5.07 if this comparison of mean returns is just one candidate out of thousands that might have produced “interesting” results.
- For example if I take a different set sample period and run the test again, or I split the sample into 5 sub-samples and run the test again.

## Traditional hypothesis testing and false discovery theory

- A primary goal of empirical research is to identify the variables involved in a phenomenon
- The identification of these variables is a prerequisite to the formulation of a theory
- In classical statistics (e.g., Econometric), the significance of variables is established through p values
-  p values suffer from multiple flaws, which have led to the acknowledgment that [most discoveries in finance are false](http://dx.doi.org/10.1111/jofi.12530)
- False discovery rate theory is an impressive advance in statistical inference, incorporating Bayesian ,frequentist, and empirical Bayesian elements.
- It was a *necessary* advance in a scientific world where computer-based technology routinely presents thousands of comparisons to be evaluated at once.


## Simulated example of false discovery problem

```{r,echo=TRUE}
set.seed(1235)
res<-tibble(testno=1:20, pvalue=1)
for (i in 1:20) {
  Up=rnorm(1000)
  NoUp=rnorm(1000)
  res[i,2]<-t.test(Up,NoUp)['p.value']
}
```
- This code fakes the return up versus down question earlier, where the *ground truth* is that there is no relationship.
- This implies that our significant results, when using a t-test, must be fake
- We repeat the test 20 times using a simple for loop.
```{r eval=require('DT')}
DT::datatable(
  res,
  fillContainer = FALSE, options = list(pageLength = 5,order = list(2, 'asc'))
)
```
- Arranging the results in ascending order of p-values.
- With enough hypothesis test we will always discovery a fake results.
- In financial machine learning, where we have to work with high-dimensional data with many features, 1000's of comparisons are common. 
